Podcast Beta
Questions and Answers
What is the focus of natural deduction in logic?
In propositional logic, what do connectives help us do?
What is the purpose of the 'Assumption' rule in natural deduction?
What happens to the proposition 'A AND B' in propositional logic if 'A' is false?
Signup and view all the answers
What does the Conjunction Elimination rule allow us to derive from A âˆ§ B?
Signup and view all the answers
In propositional logic, what does Disjunction Elimination allow us to derive?
Signup and view all the answers
Which strategy involves assuming the negation of the conclusion and deriving a contradiction?
Signup and view all the answers
What can be derived using Conditional Elimination in propositional logic?
Signup and view all the answers
In natural deduction, what is the purpose of DisjunctionTautology Elimination?
Signup and view all the answers
What rule allows us to derive 'A â‡’ B' from 'NOT A âŠ¦ B' or 'NOT(NOT B âˆ§ A)'?
Signup and view all the answers
Study Notes
Natural Deduction System: A Deep Dive into Propositional Logic, Inference Rules, and More
Natural deduction is a powerful and intuitive system for proving statements in logic, with a focus on clearly displaying the reasoning behind each conclusion. In this article, we'll explore the beauty of natural deduction as it applies to propositional logic and some of its essential components: inference rules, proof strategies, and assumptions.
Propositional Logic
Propositional logic is the foundation of natural deduction, where we reason with atomic propositions, connectives (like AND, OR, NOT, and IMPLIES), and logical operators. Propositions are statements that can be true or false, and connectives help us build more complex propositions from simpler ones.
For example, let's look at the proposition A AND B
:
 If
A
is true, andB
is true, thenA AND B
is true.  If
A
is false, orB
is false, thenA AND B
is false.
In natural deduction, we use symbols like p
, q
, r
, etc., to represent propositions, and we'll refer to these as atomic propositions.
Inference Rules
Inference rules are the fundamental building blocks of natural deduction. They dictate how we can construct valid arguments and determine the truth of propositions. Natural deduction employs a subset of inference rules, which includes:
 Assumption (âŠ¦ A): This rule allows us to temporarily assume that a statement is true for the purpose of a derivation.

Conjunction Elimination (A âˆ§ B âŠ¦ A, A âˆ§ B âŠ¦ B): This rule allows us to derive either
A
orB
fromA âˆ§ B
. 
Disjunction Elimination (A âˆ¨ B, NOT A âŠ¦ B): This rule allows us to derive
B
fromA âˆ¨ B
andNOT A
. 
Conditional Elimination (A â‡’ B, A âŠ¦ B): This rule allows us to derive
B
fromA â‡’ B
andA
. 
Negation Elimination (NOT A âŠ¦ A â‡’ B, A âŠ¦ NOT(NOT B âˆ§ A)): This rule allows us to derive
A â‡’ B
fromNOT A âŠ¦ B
orNOT(NOT B âˆ§ A)
(the absurdity form).
Proof Strategies
Natural deduction is both direct and indirect, following a collection of strategies to construct proofs.
 Direct Proof: This strategy involves using inference rules to derive the conclusion from the given assumptions.
For example, let's prove the proposition (A âˆ§ NOT B) â‡’ NOT (A âˆ¨ B)
using direct proof:
1. A âˆ§ NOT B (Assumption)
2. A (1, âˆ§Elimination)
3. NOT A â‡’ NOT (A âˆ¨ B) (Assumption)
4. A âˆ¨ B (Assumption)
5. A (4, âˆ¨Elimination) (Subproof)
6. NOT A (1, âˆ§Elimination)
7. NOT (A âˆ¨ B) (5, 6, 3, â‡’Elimination)
8. NOT (A âˆ¨ B) (27, DisjunctionTautology Elimination)
Proof:
1. A âˆ§ NOT B (Assumption)
2. A (1, âˆ§Elimination)
3. A âˆ¨ B (Assumption)
4. A (34, âˆ¨Elimination)
5. NOT A (2, âˆ§Elimination)
6. NOT (A âˆ¨ B) (4, 5, 3, â‡’Elimination)
7. NOT (A âˆ¨ B) (26, DisjunctionTautology Elimination)
QED: (A âˆ§ NOT B) â‡’ NOT (A âˆ¨ B)
 Indirect Proof (Proof by Contradiction): This strategy involves assuming the negation of the conclusion and then deriving a contradiction to conclude that the original statement is true.
For example, let's prove the proposition âˆ€x (A(x) âˆ¨ B(x)) â‡’ âˆƒx A(x) âˆ§ âˆƒx B(x)
using indirect proof:
1. âˆ€x (A(x) âˆ¨ B(x)) (Assumption)
2. Â¬(âˆƒx A(x) âˆ§ âˆƒx B(x)) (Assumption)
3. Â¬âˆƒx A(x) âˆ§ Â¬âˆƒx B(x) (2, âˆƒElimination)
4. Â¬âˆƒx A(x) (3, âˆ§Elimination)
5. âˆ€x A(x) â‡’ NOT A(t) (Assumption) for arbitrary t
6. âˆƒx (A(x) âˆ§ NOT A(t)) (4, Â¬âˆƒElimination)
7. A(s) âˆ§ NOT A(t) (6, âˆƒElimination)
8. A(s) (7, âˆ§Elimination)
9. A(s) âˆ¨ B(s) (1, UniversalInstantiation)
10. B(s) (8, 9, â‡’Elimination)
11. âˆƒx B(x) (10, âˆƒIntroduction)
12. A(t) (5, 4, âˆ€Elimination)
13. NOT A(t) (7, âˆ§Elimination)
14. Â¬A(t) (5, 12, â‡’Elimination)
15. âˆ€x A
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Explore the fundamentals of natural deduction in propositional logic, including inference rules like Assumption, Conjunction Elimination, Disjunction Elimination, Conditional Elimination, and Negation Elimination. Delve into proof strategies such as Direct Proof and Indirect Proof (Proof by Contradiction) to construct valid arguments and derive conclusions.